Rolling Cylinders

Here, we have a set of two cylinders of same size. Allow the two cylinders to roll down from the top of the incline. Notice that the two cylinders do not finish at the same time. One of the cylinders is a solid one and that is the one which finishes first. The other cylinder is a hollow one and it always finishes second.

Rolling Cylinders.jpg

In this model there are two cylinders of same size that roll down along inclined, parallel rails.

When both the cylinders are simultaneously released from the same height, we expect both to reach the bottom simultaneously.  But what we observe is that one particular cylinder always reaches the bottom well before the other.

Why should there be a time different between the two?  It is because their sizes are same but masses are not. 

One of the cylinders is hollow where as the other is a solid.

When they roll down along inclined parallel rails, there is a loss in potential energy.  As per the law of conservation of energy, this loss in potential energy has to be transformed into some other form.  Here it is converted into kinetic energy.

When the cylinder rolls down, there are two kinds of motion (i) rotation about its axis (ii) forward motion down the rails.

So, the total kinetic energy = change in potential energy = kinetic energy of rotation + kinetic energy of forward motion

what portion of the potential energy is used up for rotation and what portion is available to make use for the forward motion is decided by the rotational inertial associated with the cylinder.

More is the rotational inertia, more will be the energy used up for rotation.  So, less will be the forward motion and hence more will the time taken to reach the bottom.

What factors decide the rotational inertia of the cylinder rotating about its axis?

1. Mass     2.   Radius

Here, the radii of the two cylinders are the same but masses are different.

The solid cylinder has more mass than the hollow cylinder of same size and same material.  Therefore, its rotational inertia will be more.  (It means that it is more difficult to set it into rotation and also to stop its rotation).

So, more portion of the potential energy will be used up for rotation and less will be the portion available for forward motion.

Therefore, it takes more time to reach the bottom when compared to the hollow cylinder.